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Natural Boundaries and Integral Moments of L-Functions

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Multiple Dirichlet Series, L-functions and Automorphic Forms

Part of the book series: Progress in Mathematics ((PM,volume 300))

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Abstract

It is shown, under some expected technical assumption, that a large class of multiple Dirichlet series which arise in the study of moments of L-functions have natural boundaries. As a remedy, we consider a new class of multiple Dirichlet series whose elements have nice properties: a functional equation and meromorphic continuation. This class suggests a notion of integral moments of L-functions.

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Notes

  1. 1.

    The Poincaré series P(z, φ) is notsquare integrable. Just after an obvious Eisenstein series is subtracted, the remaining part is not only in L 2but also has sufficient decay so that its integrals against Eisenstein series converge absolutely (see [7–9]).

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Adrian Diaconu & Paul Garrett

  2. Goldfeld, Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Dorian Goldfeld

Authors
  1. Adrian Diaconu
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  2. Paul Garrett
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  3. Dorian Goldfeld
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Corresponding author

Correspondence to Adrian Diaconu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, 94305-2125, California, USA

    Daniel Bump

  2. , Department of Mathematics, Boston College, Commonwealth Ave. 140, Chestnut Hill, 02467-3806, Massachusetts, USA

    Solomon Friedberg

  3. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

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Diaconu, A., Garrett, P., Goldfeld, D. (2012). Natural Boundaries and Integral Moments of L-Functions. In: Bump, D., Friedberg, S., Goldfeld, D. (eds) Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol 300. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8334-4_7

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